1. Lecture 25: Equilibrium of a Rigid Body
ENGI 1313 Mechanics I
Lecture 25: Equilibrium of a Rigid Body

2. Lecture Objective
to illustrate application of 2D equations of equilibrium for a rigid body
to examine concepts for analyzing equilibrium of a rigid body in 3D

3. Example 25-01
Determine the force P needed to pull the 50-kg roller over the smooth step. Take θ = 60°.
 =

4. Example 25-01 (cont.) What XY-coordinate System be Established? y x
 =

5. Example 25-01 (cont.) Establish FBD y x  NB  = NA
w = mg = (50 kg)(9.807 m/s2) = 490 N

6. Example 25-01 (cont.) Determine Force Angles Roller self-weight y y x x 
70
 = 20
 = NB
 = 20 NA
w = 490 N

7. Example 25-01 (cont.) Determine Force Angles
Normal reaction force at A y x y x 
90
 = NB NA
w = 490 N NA

8. Example 25-01 (cont.) Determine Force Angles
Normal reaction force at B y x y x 
r = 0.6 m
yB = (0.6 m – 0.1 m) = 0.5 m NB
 = NB NA
w = 490 N

9. Example 25-01 (cont.) Draw FBD y y x x w = 490 N P  = 20  = 60 NB
 = NB NA
w = 490 N
NA= 0 N

10. Example (cont.)
What Equilibrium Equation should be Used to Find P?
MB = 0
w = 490 N
 = 20 P x y
 = 60
xB = m
yB = 0.5 m NB  NA

11. Comprehension Quiz 25-01
If a support prevents rotation of a body about an axis, then the support exerts a ________ on the body about that axis.
A) Couple moment
B) Force
C) Both A and B
D) None of the above.
Answer: A

12. 3-D Equilibrium Basic Equations
Moment equations can also be determined about any point on the rigid body. Typically the point selected is where the most unknown forces are applied. This procedure helps to simplify the solution.

13. Application to 3D Structures (cont.)
Engineering Design
Basic analysis
Check more rigorous methods

14. Application to 3D Structures (cont.)
Axial Forces
Design of Experimental Test Frame
Lateral Loads
Couple Forces
For Bending

15. 3-D Structural Connections
Ball and Socket
Three orthogonal forces

16. 3-D Structural Connections (cont.)
Single Journal Bearing
Two forces and two couple moments
Frictionless
Circular shaft
Orthogonal to longitudinal bearing axis

17. 3-D Structural Connections (cont.)
Journal Bearing (cont.)
Two or more (properly aligned) journal bearings will generate only support reaction forces

18. 3-D Structural Connections (cont.)
Single Hinge
Three orthogonal forces
Two couple moments orthogonal to hinge axis

19. 3-D Structural Connections (cont.)
Hinge Design
Two or more (properly aligned) hinges will generate only support reaction forces

20. Rigid Body Constraints
What is the Common Characteristic?
Statically determinate system

21. Redundant Constraints
Statically Indeterminate System
Support reactions > equilibrium equations

22. Improper Constraints Rigid Body Instability 2-D problem
Concurrent reaction forces
Intersects an out-of-plane axis

23. Improper Constraints (cont.)
Rigid Body Instability
3-D problem
Support reactions intersect a common axis

24. Improper Constraints (cont.)
Rigid Body Instability
Parallel reaction forces

25. References Hibbeler (2007)